Effective string description of the interquark potential in the 3D - - PowerPoint PPT Presentation

Effective string description of the interquark potential in the 3D U(1) Lattice Gauge Theory.1

Davide Vadacchino 1 Michele Caselle 1 Marco Panero 2 Roberto Pellegrini 1

1Universit` a degli Studi di Torino/INFN Sezion di Torino 2Instituto de F´ ısica Te´

• rica UAM/CSIC, Universidad Aut´
• noma de Madrid

Lattice 2014 Columbia University, NYC

1Caselle, Panero, Pellegrini, and Vadacchino (2014)

Outline

◮ The 3D U(1) lattice gauge model:

• 1. Confinement.
• 2. Dual formulation and gauge/string duality.

◮ Numerical results on the interquark potential: deviations with respect to

Nambu-Goto.

◮ Conclusions and future directions.

The U(1) lattice gauge model in 3D

Definition

S = β

• x∈Λ
• 1≤µ<ν≤3
• 1 − Re Ux,µUx+a ˆ

µ,νU⋆ x+aˆ ν,µU⋆ x,ν

• where Λ is a 3D euclidean spacetime lattice and

Ux,µ = exp [iaϑµ (x + aˆ µ/2)] ∈ U(1) Since the model is abelian Re Ux,µUx+a ˆ

µ,νU⋆ x+aˆ ν,µU⋆ x,ν = cos (∆µϑx,ν − ∆νϑx,µ) = cos ϑx,µν

Adopting discrete differential forms notation Z =

• c1

π

−π

d(ϑ) e−β

c2 (1−cos dϑ)

with c1 and c2 links and plaquettes on Λ.

The U(1) lattice gauge model in 3D

The weak coupling approximation

Taking the periodicity of Z into account in the β ≫ 1 approximation Z = ZswZtop = Zsw

• {q}

e−2π2β(q,∆−1q) where Ztop describes a coulomb like gas of topological excitations, Zsw describes spin-waves.

◮ The model is always confining in 3D2 ◮ In the semiclassical approximation

m0 = c0

• 8π2βe−π2βv(0),

σ ≥ cσ

• 2π2β

e−π2βv(0),

v(0) = 0.2527

the bounds are saturated and cσ = 8, c0 = 1.

◮ The ratio

m0 √σ = 2πc0 √cσ (2πβ)3/4e−π2v(0)β/2, can be tuned at will by an appropriate choice of β, in contrast to the general Yang-Mills case.

2(G¨

• pfert and Mack, 1981, Polyakov, 1977)

The U(1) lattice gauge model in 3D

The dual formulation of the model

The dual model is a globally Z symmetric spin model3 Z =

{∞}

• {⋆l=−∞}
• ⋆c1

I| d⋆l|(β), where

◮ Iα Bessel functions of order α ◮ ⋆c1 are links of the dual lattice ⋆Λ. ◮ ⋆l is an integer valued scalar field, and d⋆l differences at neighboring dual sites.

The advantage is twofold:

◮ Physical insight into the confinement mechanism : dual superconductor scenario. ◮ Ease in numerical computation. 3(Savit, 1980)

The U(1) lattice gauge model in 3D

The confinement mechanism and gauge/string duality

The dual superconductor scenario of confinement4:

◮ Condensation of magnetic monopoles drives confinement of electric charges. ◮ The dynamics of flux tubes should be described by string like degrees of freedom:

no proven gauge/string duality in the general case. In the U(1) LGT, however, an heuristic proof exists5 SPol = c1e2m0

• d2ξ√g

NG

+ c2 e2 m0

• d2ξ√gK 2

Rigidity

where c1 and c2 are two undetermined constants.

◮ At tree level, the rigidity term doens’t contribute to the interquark potential. ◮ If c1 = σ and c2 = α then

• σ/α = m ∼ m0 .

and the rigidity correction is dominant in the β → ∞ limit.

4(Polyakov, 1977) 5(Antonov, 1998, Polyakov, 1997)

The U(1) lattice gauge model in 3D

Inclusion of Polyakov lines in the partition function

The interquark potential V (R) can be extracted from G(R) = P⋆(R)P(0) = e−NtV (R) ∝

• [DX]e−Seff[X]

where Seff is the effective string action and P(x) Polykakov lines. In the dual formulation, Polyakov lines P(x) are easily included in Z L x L+ x + R L− ZR = e−βNl

{∞}

• {⋆l=−∞}
• ⋆c1

I| d⋆l+⋆n|(β) where ⋆n is integer valued and nonvanishing only on links dual to a surface bounded by the lines. Thus in the dual formulation G(R) = ZR Z which, however, is hard to measure because of an exponentially decaying signal-to-noise ratio.

Simulation of the The U(1) LGT in 3D

Snake algorithm and hierarchical update

The problem can be circumvented using the snake algorithm6: L R R + 1 G(R + 1) G(R) = ZR+1 ZR = ZR+1 Z Ld −1

R

Z Ld −1

R

Z Ld −2

R

· · · Z 1

R

ZR where Z Ld −i+1

R

Z Ld −i

R

=

• I| d∗l+1|(β)

I| d∗l|(β)

• R,Ld −i

are Ld independent local observables. And efficiency of the computation can be improved by updating the lattice hierarchically around the local observable

6(de Forcrand, D’Elia, and Pepe (2001))

Simulation of the U(1) LGT in 3D

The general setting and the measured quantity

We obtained high precision estimates of Q(R) = − 1 Nt log G(R + 1) G(R) = V (R + 1) − V (R)

◮ The dual model was simulated at several values of β on lattices L2xNt chosen to

avoid finite size effects: Nt, L =

• 64a, for β < 2.4

128a, for β ≥ 2.4

◮ Q(R) was probed in the range 1/√σ < R < L/2 ◮ A single site metropolis update algorithm was used with Jacknife error estimation.

Simulation of the U(1) LGT in 3D

Preliminary measurements

The data was fitted asymptotically with VNG (R) = σR

• 1 −

π 12σR2 using σ as free parameter in the range [Rmina, La/2]. β σa2 L, Nt 1/√σ Rmin √σ 1.7 0.122764(2) 64 3a 11a 1.9 0.066824(6) 64 4a 17a 2.0 0.049364(2) 64 5a 20a 2.2 0.027322(2) 64 6a 26a 2.4 0.015456(7) 128 8a 34a

◮ At low β, NG describes the data for a wide range of Ra ◮ As β grows, the deviations from NG grow: at β = 2.2 only 6 degrees of freedom

can be fitted! Deviations should be detectable in the range

• a/√σ, Rmina

Simulation of the U(1) LGT in 3D

Deviations with respect to NG

Deviations (Q(R) − QNG (R))a with respect to NG at β = 2.2 on a (64a)3 lattice.

1 2 3 4 5 6 R

pσ

0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 (Q(R)−QNG(R))a

The best fit value of σa2 = 0.027322(2) was obtained with Rmin √σ = 4.3.

Deviations with respect to NG

How to explain them?

In general Seff = SNG + Sb + S2,K Up to the resolution of our data SNG ≃ Scl. + σ 2

• d2ξ
• ∂αX · ∂αX − 1

4 (∂αX · ∂αX)2

• ,

Sb ≃ b2

• dξ0 [∂1∂0X · ∂1∂0X] ,

S2,K ≃ α

• (∆X)2

For each we can compute the L.O. contribution to V (R) perturbatively7 Vb(R) = −b2 π3 60 1 R4 , Vr(R) = − m 2π

∞

• n=1

K1(2nmR) n , m = σ 2α

7(Aharony and Field, 2011, Bill´

• et al., 2012, Klassen and Melzer, 1991, Nesterenko and Pirozhenko, 1997)

Deviations with respect to NG

Rigidity and boundary at LO

The boundary correction Vb alone can’t describe the deviations:

◮ χ2

R ∼ 1 only for very large values of Rmin

√σ.

◮ The best fit values of b2 have the wrong scaling behaviour:

b2σ3/2 = 0.033(3), β = 1.7 b2σ3/2 = 0.62(6), β = 2.4

◮ A complementary test with the potential

V (R) = A RB with A, B free parameters shows that b = 4. Fitting with the rigidity correction Vr works much better:

◮ Good fits are obtained already at small distances:

ma = 0.112(2), χ2

r = 1.03, Rmin

√σ = 2.15 to be compared with Rmin √σ = 4.3 for NG.

◮ The best fit value of m scales with m0.

Deviations with respect to NG

Rigidity at NLO

The NLO correction due to the rigidity contribution can be computed in the large D limit8 V2(R) = − πD 24 2 3 20mσR4 and in the general case9 V ′

2(R) = −(D − 2)(D − 10)

π 24 2 3 20mσR4

◮ This contribution is detected within the precision of our data and contributes to

the best fit value of ma.

◮ It is entangled to the boundary correction, which then cannot be neglected! 8(Braaten et al., 1987) 9(German and Kleinert, 1989)

Deviations with respect to NG

3 parameters fit of the data

V (R) = VNG (R) + Vr(R) + V ′

2(R) + Vb(R)

using σ, m and b2 as free parameters results in the best fit values σa2 = 0.027318(2), ma = 0.11(1), b2σ3/2 = 0.005(1), with χ2

r = 1.2 and Rmin

√σ = 1.65.

1 2 3 4 5 R

pσ

0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 (Q(R)−QNG(R))a

In the plot: The deviations (Q(R) − QNG (R))a and the curve Qr(R) + Q′

2(R) + Qb(R) calculated with the best fit values for σ, m and b2.

Determination of ma

The same analysis for the other couplings leads to: β ma m0a m/m0 1.7 0.28(9) 0.88(1) 0.32(10) 1.9 0.25(4) 0.56(1) 0.45(7) 2.0 0.17(2) 0.44(1) 0.39(4) 2.2 0.11(1) 0.27(1) 0.41(4) 2.4 0.06(2) 0.20(1) 0.30(10)

◮ Takes into account the interplay between σ, m, and b2 in the error. ◮ m scales with m0 as predicted by Polyakov.

Our estimate of the rigidity parameter is m/m0 = 0.35(10) .

Conclusions

◮ The strong deviations with respect to NG observed in the U(1) LGT in 3D can

be explained by the addition of a rigidity term to the effective string action, as predicted by Polyakov. This contribution becomes dominant in the limit β → ∞.

◮ Future directions:

• 1. Try to disentangle the NLO rigidity contribution from the boundary correction.
• 2. Study the behaviour intrinsic width of the string and compare with predictions of the

string with rigidity.

• 3. Finite temperature behaviour of the interquark potential.

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